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Chapter 0: Problem 75

Apply the distributive property. $$ 3\left(4 p^{3}-6.1 p^{2}-8 p+2.2\right) $$

Short Answer

Expert verified
12p^3 - 18.3p^2 - 24p + 6.6

Step by step solution

01

- Understand the Distributive Property

The distributive property states that for any numbers or expressions a, b, and c: \( a(b + c) = ab + ac \). This means you need to multiply each term inside the parenthesis by the number outside, which in this case is 3.
02

- Multiply Each Term

Multiply each term inside the parenthesis by 3: \[ 3 \times 4p^3 = 12p^3 \] \[ 3 \times (-6.1p^2) = -18.3p^2 \] \[ 3 \times (-8p) = -24p \] \[ 3 \times 2.2 = 6.6 \]
03

- Combine the Results

After performing the multiplications, combine the results to form the final expression as: \[ 12p^3 - 18.3p^2 - 24p + 6.6 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Multiplication
Polynomial multiplication involves multiplying a polynomial by another polynomial or a constant. In our exercise, we are multiplying a polynomial by the constant 3. To do this, we need to multiply each term in the polynomial by 3 and then combine the resulting terms. By distributing the constant across each term in the polynomial, we ensure that every term is multiplied correctly.
Algebraic Expressions
Algebraic expressions are a mix of numbers, variables, and mathematical operations (like addition and subtraction). For example, in the polynomial 4p^3 - 6.1p^2 - 8p + 2.2, we have:
  • The term 4p^3 (a number along with a variable raised to a power)
  • The term -6.1p^2 (a negative number with a variable and an exponent)
  • The term -8p (a negative number and a variable)
  • The term 2.2 (just a constant number)

Understanding how to handle each term is important, especially when applying the distributive property or other algebraic operations.
Distributive Property
The distributive property is a key concept in algebra. It allows us to simplify expressions and solve equations more easily. The property states that: \(a(b + c) = ab + ac\). This means we multiply each term inside the parenthesis by the factor outside. In our example, we used this property to multiply 3 with each term in the polynomial 4p^3 - 6.1p^2 - 8p + 2.2. Here are the steps once more:
  • Multiply 3 by 4p^3 to get 12p^3
  • Multiply 3 by -6.1p^2 to get -18.3p^2
  • Multiply 3 by -8p to get -24p
  • Multiply 3 by 2.2 to get 6.6

Combining these results gives us the final expression: 12p^3 - 18.3p^2 - 24p + 6.6. This property helps to break down complex expressions into simpler parts that are easier to manage.

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Most popular questions from this chapter

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (2 v-\sqrt{17})^{2} $$

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